Let V be an n-dimensional vector space with an ordered basis B.
Define T: V → F^n by T(x) = [x]B. Prove that T is linear.
Note that the [x]B is suppose to be a subscript B
You just need to show: (i) $\displaystyle T(k\bold{v}) = kT(\bold{v})$ i.e. $\displaystyle [k\bold{v}]_B = k[\bold{v}]_B$ (ii)$\displaystyle T(\bold{v}+\bold{w}) = T(\bold{v}) + T(\bold{w})$ i.e. $\displaystyle [\bold{v}+\bold{w}]_B = [\bold{v}]_B + [\bold{w}]_B$.